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Exam1 Review

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					Exam1 Review

   Dr. Bernard Chen Ph.D.
  University of Central Arkansas
           Spring 2010
Computer Science at a
Crossroads
“Power wall”
 Triple hurdles of maximum power dissipation of air-
  cooled chips

“ILP wall”
 Little instruction-level parallelism left to exploit
   efficiently

“Memory wall”
 Almost unchanged memory latency
Computer Science at a
Crossroads
   Old Conventional Wisdom : Uniprocessor
    performance 2X / 1.5 yrs

   New Conventional Wisdom :
    Power Wall + ILP Wall + Memory Wall =
    Brick Wall

       Uniprocessor performance now 2X / 5(?) yrs
                               Computer Science at a
                               Crossroads
                               10000


                                                                                                 ??%/year
                                1000
Performance (vs. VAX-11/780)




                                                                  52%/year

                                 100




                                  10
                                            25%/year



                                   1
                                   1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Defining Computer
Architecture
   The task of computer designer:
    Determine what attributes are
    important for a new computer, then
    design a computer to maximize
    performance while staying within
    cost, power, and availability constrains
Defining Computer
Architecture
   In the past, the term computer architecture often
    referred only to instruction set design
   Other aspects of computer design were called
    implementation, often assuming that implementation
    is uninteresting or less challenging

   Of course, it is wrong for today’s trend
   Architect’s job much more than instruction set
    design; technical hurdles today more challenging
    than those in instruction set design
Instruction Set Architecture
(ISA)
   The instruction set architecture serves
    as the boundary between the software
    and hardware.

   We will have a complete introduction to
    this part. (Some examples in the next
    two slides)
Outline
   Decoder
   Encoder
   MUX
2-to-4 Decoder
Decoder Expansion
How about 4-16 decoder
   Use how many 3-8 decoder?
   Use how many 2-4 decoder?
Encoders

   Perform the inverse operation of a
    decoder
      2 (or less) input lines and n output
         n

       lines
Encoders
Priority Encoder


   Accepts multiple values and encodes them
      Works when more than one input is active

   Consists of:
      Inputs (2 )
                 n

      Outputs

          when more than one output is active, sets
           output to correspond to highest input
          V (indicates whether any of the inputs are
           active)
      Selectors / Enable (active high or active low)
          4 to 1 line multiplexer




4 to 1 line
multiplexer
2n MUX to 1                 S1   S0   F
                            0    0    I0
n for this MUX is 2         0    1    I1

This means 2                1    0    I2
                            1    1    I3
selection lines s0
and s1
Outline
   Data Representation
   Compliments
     Conversion Between Number Bases
                                           Octal(base 8)


Decimal(base 10)      Binary(base 2)



                                           Hexadecimal
                                             (base16)
 °   We normally convert to base 10
 because we are naturally used to the
    decimal number system.
 °   We can also convert to other number
     systems
Example
 Convert 101011110110011 to
a.   octal number
b. hexadecimal number

a.    Each 3 bits are converted to octal :
                             (101) (011) (110) (110) (011)
                                                   
                               5      3    6     6     3
               101011110110011 = (53663)8
 b. Each 4 bits are converted to hexadecimal:
                             (0101) (0111) (1011) (0011)
                                                
                                5     7      B     3
               101011110110011 = (57B3)16
 Conversion from binary to hexadecimal is similar except that the bits
divided into groups of four.
     Subtraction using addition
• Conventional addition (using carry) is easily
• implemented in digital computers.
• However; subtraction by borrowing is difficult
  and inefficient for digital computers.
• Much more efficient to implement subtraction
  using ADDITION OF the COMPLEMENTS of
  numbers.
       Complements of numbers
(r-1 )’s Complement
•Given a number N in base r having n digits,
•the (r- 1)’s complement of N is defined as
                (rn - 1) - N


•For decimal numbers the
base or r = 10 and r- 1= 9,         9       9       9       9       9
•so the 9’s complement of N
is      (10n-1)-N               -   Digit
                                    n
                                            Digit
                                            n-1
                                                    Next
                                                    digit
                                                            Next
                                                            digit
                                                                    First
                                                                    digit


•99999……. - N
l’s complement
   For binary numbers, r = 2 and r — 1 =
    1,
   r-1’s complement is the l’s complement.
   The l’s complement of N is (2^n- 1) - N.

            Bit n-1   Bit n-2   …….     Bit 1   Bit 0

            1         1         1       1       1


        -   Digit
            n
                      Digit
                      n-1
                                Next
                                digit
                                        Next
                                        digit
                                                First
                                                digit
     r’s Complement
•Given a number N in base r having n digits,
•the r’s complement of N is defined as
               rn - N.

•For decimal numbers the
base or r = 10,
                                1       0       0       0       0       0
•so the 10’s complement of N
is      10n-N.                      -   Digit
                                        n
                                                Digit
                                                n-1
                                                        Next
                                                        digit
                                                                Next
                                                                digit
                                                                        First
                                                                        digit

•100000……. - N
  10’s complement Examples
Find the 10’s complement of
546700 and 12389                    1   0   0   0   0   0   0


The 10’s complement of 546700       -   5   4   6   7   0   0

is 1000000 - 546700= 453300
                                        4   5   3   3   0   0
and the 10’s complement of
12389 is                                1   0   0   0   0   0

100000 - 12389 = 87611.             -       1   2   3   8   9

Notice that it is the same as 9’s           8   7   6   1   1
complement + 1.
     2’s complement

For binary numbers, r = 2,
r’s complement is the 2’s complement.
The 2’s complement of N is 2n - N.


                                 1       0       0       0       0       0


                                     -   Digit
                                         n
                                                 Digit
                                                 n-1
                                                         Next
                                                         digit
                                                                 Next
                                                                 digit
                                                                         First
                                                                         digit
Subtraction of Unsigned Numbers
using r’s complement
   (1) if M  N, ignore the carry without
    taking complement of sum.
    (2) if M < N, take the r’s complement
    of sum and place negative sign in front
    of sum. The answer is negative.
Subtract by Summation
   Subtraction with complement is done
    with binary numbers in a similar way.

   Using two binary numbers X=1010100
    and Y=1000011

   We perform X-Y and Y-X
X-Y
   X=                1010100
   2’s com. of Y=    0111101
   Sum=             10010001
   Answer=           0010001
Y-X
   Y=               1000011
   2’s com. of X=   0101100
   Sum=             1101111

   There’s no end carry: answer is
    negative --- 0010001 (2’s complement
    of 1101111)
How To Represent Signed
Numbers
    Plus and minus signs used for decimal
     numbers: 25 (or +25), -16, etc.

    For computers, it is desirable to represent
     everything as bits.

    Three types of signed binary number
     representations:
1.   signed magnitude,
2.   1’s complement, and
3.   2’s complement
      1. signed magnitude
      • In each case: left-most bit
        indicates sign: positive (0) or
        negative (1).

Consider 1. signed magnitude:

           000011002 = 1210              100011002 = -1210

Sign bit       Magnitude      Sign bit       Magnitude
    2. One’s Complement
    Representation
        The one’s complement of a binary
         number involves inverting all bits.

     • To find negative of 1’s complement number
         take the 1’s complement of whole number
         including the sign bit.

      000011002 = 1210          111100112 = -1210

Sign bit    Magnitude     Sign bit   1’complement
3. Two’s Complement
Representation
• The two’s complement of a binary
  number involves inverting all bits and
  adding 1.
 To find the negative of a signed number

  take the 2’s the 2’s complement of the
  positive number including the sign bit.

      000011002 = 1210         111101002 = -1210

Sign bit   Magnitude     Sign bit   2’s complement
    Sign addition in 2’s complement
The rule for addition is add the two numbers, including their sign bits,
and discard any carry out of the sign (leftmost) bit position.
Numerical examples for addition are shown below.
Example:
       +6        00000110 - 6 11111010
       +13       00001101 +13 00001101
       +19       00010011 +7          00000111

                   +6        00000110 -6         11111010
                   -13       11110011     -13    11110011
       -7        11111001      -19     11101101
In each of the four cases, the operation performed is always addition,
including the sign bits.
Only one rule for addition, no separate treatment of subtraction.
Negative numbers are always represented in 2’s complement.
Overflow
   Overflow example:

    +70   0 1000110        -70   1 0111010
    +80   0 1010000        -80   1 0110000

= +150 1 0010110        =-150 0 1101010

   An overflow may occur if the two numbers added are
    both either positive or negative.
BINARY ADDER-SUBTRACTOR
     Example
    Extend the previous logic circuit to accommodate XNOR,
     NAND, NOR, and the complement of the second input.

S2   S1   S0   Output      Operation
0    0    0    XY         AND
0    0    1    XY         OR
0    1    0    XY         XOR
0    1    1    A           Complement A
1    0    0    (X  Y)     NAND
1    0    1    (X  Y)     NOR
1    1    0    (X  Y)     XNOR
1    1    1    B           Complement B
Shift Microoperations
Symbolic designation                 Description
    R ← shl R                Shift-left register R
    R ← shr R                Shift-right register R
    R ← cil R                Circular shift-left register R
    R ← cir R                Circular shift-right register R
    R ← ashl R               Arithmetic shift-left R
    R ← ashr R               Arithmetic shift-right R

            TABLE 4-7. Shift Microoperations
     Logical Shift Example

1. Logical shift: Transfers 0 through the serial input.
R1 shl R1 Logical shift-left
R2 shr R2 Logical shift-right

(Example) Logical shift-left
10100011                 01000110

(Example) Logical shift-right
10100011                 01010001
   Circular Shift Example

        Circular shift-left    R1  cil R1
        Circular shift-right   R 2  cir R 2

(Example) Circular shift-left
10100011 is shifted to 01000111
(Example) Circular shift-right
10100011 is shifted to 11010001
Arithmetic Shift Right
   Arithmetic Shift Right :
       Example 1
             0100 (4) 
             0010 (2)
       Example 2
             1010 (-6) 
             1101 (-3)
Arithmetic Shift Left
   Arithmetic Shift Left :
        Example 1
                     0010 (2) 
                     0100 (4)
        Example 2
                     1110 (-2) 
                     1100 (-4)

   Arithmetic Shift Left :
        Example 3
                     0100 (4) 
                     1000 (overflow)
        Example 4
                     1010 (-6) 
                     0100 (overflow)

				
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